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In null-hypothesis significance testing, the p-value [note 1] is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. [2] [3] A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis.
The p-value is the probability that a test statistic which is at least as extreme as the one obtained would occur under the null hypothesis. At a significance level of 0.05, a fair coin would be expected to (incorrectly) reject the null hypothesis (that it is fair) in 1 out of 20 tests on average.
There would be no special value in "creative leaps", no fundamental gap between solving a problem and recognizing the solution once it's found. — Scott Aaronson , UT Austin On the other hand, some researchers believe that it is overconfident to believe P ≠ NP and that researchers should also explore proofs of P = NP.
To determine whether a result is statistically significant, a researcher calculates a p-value, which is the probability of observing an effect of the same magnitude or more extreme given that the null hypothesis is true. [5] [12] The null hypothesis is rejected if the p-value is less than (or equal to) a predetermined level, .
There, one uses a likelihood function for all possible values of the prior instead of the p-value for a single null hypothesis. The p-value describes a property of data when compared to a specific null hypothesis; it is not a property of the hypothesis itself. For the same reason, p-values do not give the probability that the data were produced ...
If any sum of the numbers can be specified with at most P bits, then solving the problem approximately with = is equivalent to solving it exactly. Then, the polynomial time algorithm for approximate subset sum becomes an exact algorithm with running time polynomial in n and 2 P {\displaystyle 2^{P}} (i.e., exponential in P ).
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory.One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.
How to Solve It suggests the following steps when solving a mathematical problem: . First, you have to understand the problem. [2]After understanding, make a plan. [3]Carry out the plan.